Generalization of the ad hoc refinement argument

Develop a broader generalization of the ad hoc argument used to analyze imprimitive transitive permutation groups with derangement graphs having no clique of size four.

Background

In proving Lemma 4.1, the authors deliberately avoid applying Proposition 2.9 and instead use an ad hoc argument tailored to the relevant imprimitive action. The argument yields the required degree-six quotient and is an important step in reducing the exceptional cases in Theorem 1.1.

The authors explicitly state that they do not know how to extend this method to a broader setting. Because the unresolved issue concerns a concrete generalization of the proof technique, it qualifies as an open problem under the stated inclusion criteria.

References

Although we are in a position to apply Proposition 2.9, we choose not to do so here. Instead, we argue using an ad-hoc method, as we believe this approach could be applied more generally. However, at present, we do not know how to achieve this broader generalization.

— Kronecker classes and cliques in derangement graphs  (2502.01287 - Cazzola et al., 3 Feb 2025) in Section 4, proof of Lemma 4.1